Notice that the decomposition given by Equation REF  implies a second assumption; specifically, that the \(N\)  local latent variables \(\mathbf {z}_n\)  are exchangeable in their joint distribution. The joint distribution here refers to the prior \(p(\mathbf {z}_n\mid \theta )\)  which is governed by the global latent variable \(\theta \) . The role of global latent variable in the model is to control the extent of information pooling across the observations. To illustrate, if we set the variance of the prior distribution to infinity, we obtain a separate model for each group as a result. That is, a different model is fit to each observation vector, where \(\mathbf {y}_n\)  can be thought of as an “experiment” with \(P\)  measurements. On the other extreme, setting the variance to zero would result in complete pooling of information, effectively giving us a single model for all \(N \times P\)  collapsed observations [1]}. Incidentally, in the context of mixed effects models, it is interesting to point out that a fixed effect can be thought of as a special case of a random effect, but one with an infinite variance prior.
